Journal of Lie Theory Volume 10 (2000) 443–453 C ◯2000 Heldermann Verlag

Embeddings of almost homogeneous Heisenberg groups

Markus Stroppel

Communicated by K.-H. Neeb

Abstract. We study injective continuous homomorphisms between simply connected Lie groups that are nilpotent of class 2 and whose group of automorphisms acts with 3 orbits.

1. Introduction

A topological group G is called almost homogeneous if the group Aut(G) of all topological automorphisms has 3 orbits on G. In the general case, there are different constructions leading to almost homogeneous groups (including Sym(3), Alt(4), additive groups of free modules over the ring ℤ/p^{2}ℤ for a prime p, Suzuki 2-groups, semidirect products Q ⋉ V with respect to a suitable linear representation of a cyclic group Q on a vector space V , and certain generalized Heisenberg groups), compare [2].

In [5], it has been shown that every almost homogeneous locally compact connected group is a simply connected Lie group which is nilpotent of class 2: that is, a Heisenberg group. The Heisenberg groups are obtained by the following

Construction. Let V and Z be vector spaces of finite dimension over ℝ, and let β: V × V → Z be a symplectic map (that is, a skew-symmetric bilinear map). Then the vector space V × Z becomes a Lie algebra 𝔤𝔥 (V, Z, β) with Lie bracket^{†} [(v, x), (w, y)] = (0, (v, w)^{β}). The corresponding simply connected Lie group GH (V, Z, β) is then the space V × Z with (Campbell–Hausdorff) multiplication (v, x) ∗ (w, y) = (v + w, x + y + \frac{1}{2}(v, w)^{β}). If H := GH (V, Z, β) is not abelian then the set

C := {(0, (v, w)^{β}) | v, w ∈ V }

ISSN 0949–5932 / $2.50 ^{_{C} ◯} Heldermann Verlag of commutators is a nontrivial characteristic subset, and it is contained in the center of H which is a proper characteristic subset of H. In an almost homogeneous group H, we thus have that C and the center of H coincide; this entails C = {0}×Z. Note also that β is nondegenerate if GH (V, Z, β) is almost homogeneous.

† Maps will be applied from the right, and we use exponential notation. Linear maps, however, will be denoted by juxtaposition; this should remind of multiplication of row vectors by matrices.

2. Classification of Almost Homogeneous Heisenberg Groups

The almost homogeneous ones among the Heisenberg groups have been determined in [4]. They are characterized by the following result, see [4] 2.2 and 4.5.

Proposition 2.1. Let β: V × V → Z be a nonzero symplectic map. The Heisenberg group GH (V, Z, β) is almost homogeneous if, and only if, there is a compact group Φ inducing groups of orthogonal transformations on V and on Z that act transitively on the respective spheres, and such that the equation (vφ, wφ)^{β} = (v, w)^{β}φ holds for all v, w ∈ V and all φ ∈ Φ.

Transitive actions on spheres are well understood, compare [3] 96.16–24. The symplectic map β is somewhat hard to deal with. Standard methods of multi-linear algebra lead us to consider the exterior product V ∧ V (that is, the space of skew-symmetric tensors) and the linearization β̄: V ∧ V → Z of β. The action of Φ on V then yields an action of Φ on V ∧ V , and the map β̄ is a homomorphism of ℝ[Φ]-modules. The ℝ[Φ]-modules V and Z are both simple because of the transitive action of Φ on the respective sphere. As an almost homogeneous Heisenberg group GH (V, Z, β) is generated by V × {0}, we may (and will) assume that Φ acts effectively on V ; the reader should be aware, however, that this does not imply that Φ acts effectively on V ∧ V . As Φ is compact, the ℝ[Φ]-module V ∧ V is semi-simple, and the surjective homomorphism β̄ may be considered as projection onto some suitable simple submodule of V ∧ V . We will freely use the representation theory of compact groups, cf. [1].

The ℝ[Φ]-module V ∧ V has a convenient model: we identify V ∧ V with the Lie algebra 𝔬_{d} of all skew-symmetric d × d matrices over ℝ, where d = dim V . The universal bilinear map from V × V to V ∧ V is then given by mapping (v, w) to the matrix v^{tr}w − w^{tr}v, where we consider the elements of V as row vectors, and v^{tr} denotes the transpose of v. The action of the group Φ on V ∧ V is obtained by identifying Φ with a subgroup of the orthogonal group O_{d}ℝ and restricting the adjoint action of O_{d}ℝ on its Lie algebra 𝔬_{d}: that is, a matrix A ∈ Φ maps X ∈ 𝔬_{d} to A^{tr}XA = A^{−1}XA. This interpretation makes it easy to find certain submodules of the ℝ[Φ]-module 𝔬_{d}: for instance, we have the Lie algebra 𝔣 of Φ, and its centralizer (which is a submodule on which Φ acts trivially).

The results of [4] amount to a complete classification of almost homogeneous Heisenberg groups. There are three infinite series:

Examples 2.2. First of all, one has nondegenerate symplectic 𝔽-bilinear forms β_{𝔽}^{n}: 𝔽^{2n} × 𝔽^{2n} → 𝔽, for 𝔽 ∈ {ℝ, ℂ} and each positive integer n. This leads to almost homogeneous Heisenberg groups H_{ℝ}^{2n} := GH (ℝ^{2n}, ℝ, β_{ℝ}^{n}) and H_{ℂ}^{4n} := GH (ℂ^{2n}, ℂ, β_{ℂ}^{n}), respectively. The skewfield ℍ of Hamilton’s quaternions does not admit nonzero ℍ-bilinear forms. However, each anisotropic hermitian form γ_{ℍ}^{n}: ℍ^{n} × ℍ^{n} → ℍ leads to a symplectic ℝ-bilinear map β_{ℍ}^{n}: ℍ^{n} × ℍ^{n} → Pu (ℍ) via (v, w)^{β_{ℍ}^{n}} := Pu ((v, w)^{γ_{ℍ}^{n}}). Thus we obtain another infinite series of almost homogeneous Heisenberg groups H_{ℍ}^{4n} := GH (ℍ^{n}, Pu (ℍ) , β_{ℍ}^{n}).

The proof that these series consist of almost homogeneous groups uses the fact that, for 𝔽 ∈ {ℝ, ℂ}, the symplectic group Sp_{2n}𝔽 contains a compact subgroup U_{n}𝕂 (where 𝕂 is the skewfield extension of degree 2 over 𝔽) acting transitively on the unit sphere of 𝔽^{2n}, and transitivity of U_{n}ℍ on the unit sphere in ℍ^{n}. It remains to enlarge these compact groups in order to obtain transitivity on the spheres in ℝ, ℂ, and Pu (ℍ), respectively.

In the case of H_{ℝ}^{2n}, one may take a finite extension of U_{n}ℂ for the group Φ. For 𝔽 ∈ {ℂ, ℍ} one uses Φ = 𝕊_{𝔽} · U_{n}ℍ to show that H_{𝔽}^{4n} is almost homogeneous; here 𝕊_{𝔽} is the group consisting of scalar multiplications by elements of the unit sphere in 𝔽.

Apart from these infinite series, there are 6 exceptional almost homogeneous Heisenberg groups. The group H_{z}^{d} is obtained using a subgroup Φ of SO_{d}ℝ, the ℝ[Φ]-module 𝔬_{d}, and the projection onto some simple submodule Z of dimension z in 𝔬_{d}. More specifically:

Examples 2.3: Exceptional almost homogeneous Heisenberg groups.

• The group H_{3}^{3} is obtained from Φ = SO_{3}ℝ, with Z = 𝔬_{3} .

• The group H_{6}^{6} is obtained from Φ = SU_{3}ℂ, with Z the complement to the Lie algebra of U_{3}ℂ in the ℝ[SU_{3}ℂ]-module 𝔬_{6} .

• The group H_{7}^{7} is obtained from Φ = G_{2} , with Z the complement of the Lie algebra of G_{2} in the ℝ[G_{2}]-module 𝔬_{7} .

• The group H_{5}^{8} is obtained from Φ = U_{2}ℍ, with Z one of the fivedimensional submodules in the complement of the sum of the Lie algebra of U_{2}ℍ and its 3-dimensional centralizer in the ℝ[U_{2}ℍ]-module 𝔬_{8} .

• The group H_{6}^{8} is obtained from Φ = SU_{4}ℂ, with Z one of the simple 6-dimensional submodules of the 12-dimensional complement of the Lie algebra of U_{4}ℂ in the ℝ[SU_{4}ℂ]-module 𝔬_{8} .

• The group H_{7}^{8} is obtained from Φ = Spin_{7} , with Z the complement of the Lie algebra of Spin_{7} in the ℝ[Spin_{7}]-module 𝔬_{8} .

For details, see the discussion in [4]. The main result of that paper is the following (see [4] 7.5):

Theorem 2.4. Let H = GH (V, Z, β) be an almost homogeneous real Heisenberg group. Then H is isomorphic to an element of the series H_{ℝ}^{2n}, H_{ℂ}^{4n}, or H_{ℍ}^{4n}, where n runs over the positive integers; or H is isomorphic to one of the groups H_{3}^{3} , H_{6}^{6} , H_{7}^{7} , H_{5}^{8} , H_{6}^{8} , and H_{7}^{8} .

As we are interested in embeddings, the following alternative descriptions of the groups H_{3}^{3} , H_{6}^{6} , and H_{7}^{7} are helpful. For 𝔽 ∈ {ℝ, ℂ}, let V be a 3-dimensional vector space over 𝔽. Then the exterior product V ∧ V , formed over 𝔽, is a 3-dimensional vector space over 𝔽 which may be identified with the orthogonal Lie algebra 𝔬_{3}𝔽. Mapping (v, w) to v ∧ w (that is, the commutator map in 𝔬_{3}𝔽) yields a symplectic map ∧: V × V → V ∧ V and an almost homogeneous Heisenberg group S_{𝔽} := GH (V, V ∧ V, ∧). A second method to construct almost homogeneous Heisenberg groups starts with a noncommutative alternative division algebra 𝔽 over ℝ (that is, with 𝔽 ∈ {ℍ, 𝕆}). The commutator map induces a symplectic map from Pu (𝔽) × Pu (𝔽) onto Pu (𝔽), leading to almost homogeneous Heisenberg groups C_{ℍ} and C_{𝕆}, respectively. The classification 2.4 yields isomorphisms H_{3}^{3} ≅ S_{ℝ} ≅ C_{ℍ}, H_{6}^{6} ≅ S_{ℂ}, and H_{7}^{7} ≅ C_{𝕆}.

3. Embeddings

We already noticed the isomorphisms S_{ℝ} ≅ C_{ℍ} ≅ H_{3}^{3} and C_{𝕆} ≅ H_{7}^{7} . There are obvious inclusions H_{ℝ}^{2n} ≤ H_{ℂ}^{4n}, for each natural number n. Restricting ourselves to the subspace ℂ^{n} of ℍ^{n}, we also find inclusions of H_{ℝ}^{2n} in H_{ℍ}^{4n}; note that the set of commutators of elements from ℂ^{n} × ℂ^{n} is a vector space of dimension 1 since we take the pure part of a complex number. Moreover, we have an inclusion H_{3}^{3} ≅ S_{ℝ} ≤ S_{ℂ} ≅ H_{6}^{6} , and an inclusion H_{3}^{3} ≅ C_{ℍ} ≤ C_{𝕆} ≅ H_{7}^{7} . It appears reasonable to ask for other embeddings.

In general, an injective continuous group homomorphism need not be an embedding (that is, a homeomorphism onto its image). In the present situation, we are dealing with connected, simply connected, nilpotent Lie groups, where this problem does not occur: every injective continuous homomorphism between such groups is an embedding. However, we will not use that fact in the sequel, because we reduce our problems to ℝ-linear maps. Recall that continuous group homomorphisms between vector groups (that is, additive groups of vector spaces of finite dimension over ℝ) are ℝ-linear. Thus it makes sense to consider dimensions of the vector groups H′ and H/H′, for any Heisenberg group H.

There is one Heisenberg group that appears to be omni-present:

Proposition 3.1. Let G = GH (V, Z, β) be any noncommutative Heisenberg group. Then there is an embedding of H_{ℝ}^{2} into G.

Proof. We pick elements x, y ∈ G such that z := [x, y] ≠ 1. Putting S := ℝx + ℝy and Y := ℝz we obtain the subgroup GH (S, Y, β|_{S×S}) ≅ H_{ℝ}^{2} in G.

Lemma 3.2. Let G = GH (S, Y, γ) and H = GH (V, Z, β) be almost homogeneous Heisenberg groups. Assume that φ: G → H is an injective continuous homomorphism. Then the following hold.

a. The commutator groups satisfy G′^{φ} = H′ ∩ G^{φ}.

b. There are injective linear maps φ_{1}: S → V and φ_{2}: Y → Z such that the following diagram commutes:

[figure: S × S; γ; Y; V × V; (φ_{1}, φ_{1}); β; Z; φ_{2}]

c. We have inequalities dim G′ ≤ dim H′ and dim(G/G′) ≤ dim(H/H′). d. If dim(G/G′) = dim(H/H′) then φ_{1} , φ_{2} and φ are isomorphisms.

Proof. First of all, we note that G′^{φ} = (G^{φ})′ ≤ H′ yields that φ induces an injection of G′ = {0} × Y into H′ = {0} × Z. This gives the injective linear map φ_{2} from Y into Z. Moreover, we obtain an injective homomorphism from G^{φ}/(H′ ∩ G^{φ}) into H/H′. As H′ coincides with the center of H, we have that H′ ∩ G^{φ} is contained in the center of G^{φ}. But the latter is G′^{φ}, and we have proved assertion a. Moreover, we obtain an inclusion φ̄: G/G′ → H/H′, and a corresponding linear injection φ_{1}: S → V .

For each s ∈ S we write g_{s} := (s, 0) ∈ G. Then we have (G′g_{s})φ̄ = H′(s^{φ_{1}}, 0), and [g_{s}^{φ}, g_{t}^{φ}] = (0, (s^{φ_{1}}, t^{φ_{1}})^{β}) leads to (0, (s, t)^{γφ_{2}}) = [g_{s}, g_{t}]^{φ} = [g_{s}^{φ}, g_{t}^{φ}] = (0, (s^{φ_{1}}, t^{φ_{1}})^{β}). This completes the proof of assertion b. We obtain a commutative diagram

[figure: S ∧ S; G/G′ ∧ G/G′; γ̄; G′; Y; V ∧ V; φ_{1} ∧ φ_{1}; H/H′ ∧ H/H′; φ̄ ∧ φ̄; β̄; H′; φ̃; Z; φ_{2}]

where φ̃ is obtained by restriction of φ. Equality dim(G/G′) = dim(H/H′) yields that φ_{1} is an isomorphism. Then surjectivity of β̄ entails that the embedding φ_{2} is also surjective, and we obtain the last assertion.

Corollary 3.3. There are no injective continuous homomorphisms between the groups H_{ℝ}^{4n}, H_{ℂ}^{4n} and H_{ℍ}^{4n}, nor between H_{ℝ}^{6} and H_{6}^{6} ; and there are no injective continuous homomorphisms between the groups H_{ℝ}^{8} , H_{ℂ}^{8} , H_{ℍ}^{8} , H_{5}^{8} , H_{6}^{8} , and H_{7}^{8} .

Proposition 3.4. There is no injective homomorphism from H_{ℂ}^{4} into H_{ℍ}^{8} .

Proof. Assume, to the contrary, that there is an injective homomorphism φ from H_{ℂ}^{4} into H_{ℍ}^{8} . We note that φ may also be considered as an (injective) homomorphism of Lie algebras, because the exponential map is bijective (in fact, the identity) for these groups. We consider the elements

w := ((1, 0), 0) , x := ((i, 0), 0) , y := ((0, 1), 0) , and z := ((0, i), 0)

in H_{ℂ}^{4} . Our assumption that φ is injective entails that none of these elements is mapped into the center of H_{ℍ}^{8} . Since H_{ℍ}^{8} is almost homogeneous, we may assume w^{φ} = ((1, 0), 0). As w and x commute with each other, we have x^{φ} = ((ρ, σ), p) with (ρ, σ) ∈ ℝ × ℍ and p ∈ Pu (ℍ). Using a suitable element in the group U_{2}ℍ (which acts as a group of automorphisms on H_{ℍ}^{8} ), we achieve σ ∈ ℝ, as well. We write ((α + a, β + b), q) := y^{φ} and ((γ + c, δ + d), u) := z^{φ} with (α, a), (β, b), (γ, c), (δ, d) ∈ ℝ × Pu (ℍ) and q, u ∈ Pu (ℍ). The commutator relations for w, x, y, z in the Lie algebra of H_{ℂ}^{4} are translated by φ into the relations

[w^{φ}, y^{φ}] = −[x^{φ}, z^{φ}], [w^{φ}, z^{φ}] = [x^{φ}, y^{φ}], and [y^{φ}, z^{φ}] = 0.

The first two of these relations yield the equations

σd = −a − ρc and σb = c − ρa.

As φ is injective, we also have that [w^{φ}, y^{φ}] = ((0, 0,), −a) and [w^{φ}, z^{φ}] = ((0, 0,), −c) are linearly independent over ℝ. This implies σ ≠ 0, and the third commutator relation now gives

0 = Pu ((α + a)(γ − c) + (β + b)(δ − d)) ∈ ℝa + ℝc − \frac{σ^{2} + 1 + ρ^{2}}{σ^{2}} Pu (ac) .

However, since the elements a, c ∈ Pu (ℍ) are linearly independent, the three elements a, c, and Pu (ac) = a×c are linearly independent, as well. This means that the positive real number σ^{2} + 1 + ρ^{2} has to be zero, a contradiction.

Proposition 3.5. There is an embedding of H_{ℂ}^{4} into H_{ℍ}^{12} . Proof. Resuming the notation from the proof of 3.4, we notice that w^{φ} := ((1, 0, 0), 0), x^{φ} := ((0, 1, 0), 0), y^{φ} := ((i, j, 1), 0), z^{φ} := ((j, −i, −2k), 0) defines an embedding φ: H_{ℂ}^{4} → H_{ℍ}^{12} , as claimed.

Theorem 3.6. Let k and n be positive integers, let 𝔽, 𝕂 ∈ {ℝ, ℂ, ℍ} and assume that there is an injective continuous homomorphism φ: H_{𝔽}^{2k} → H_{𝕂}^{4n}. Then k > n implies 𝔽 = 𝕂. Proof. From 3.2 we know 𝔽 ≤ 𝕂. For 𝕂 = ℝ, there is nothing left to prove. Now assume k > n. We have H_{𝔽}^{2k} = GH (V, Z, β) with dim_{ℝ} V = 2k > 2n. The image of V under φ̄ will be denoted by S.

In the case 𝕂 = ℂ, we have dim_{ℝ} S > \frac{1}{2} dim_{ℝ} ℂ^{2n}, and the intersection S ∩iS is not trivial. Therefore, we find u, v ∈ V \{0} with vφ̄ = iuφ̄. Commutators [(x, 0), (y, 0)] in H_{ℝ}^{2d} or H_{ℂ}^{2d} are both given as (0, ⟨x|y⟩), where ⟨·|·⟩ denotes the (essentially unique) nondegenerate symplectic form on ℝ^{2d} and ℂ^{2d}, respectively. Picking w ∈ V such that ⟨u|w⟩ ≠ 0, we compute 0 ≠ (0, ⟨uφ̄|wφ̄⟩) = [(u, 0), (w, 0)]^{φ} and [(v, 0), (w, 0)]^{φ} = (0, ⟨vφ̄|wφ̄⟩) = (0, i⟨uφ̄|wφ̄⟩). This means that the commutator of (H_{𝔽}^{2k})^{φ} ≅ H_{𝔽}^{2k} has dimension 2, and 𝔽 = ℂ follows.

Now consider the case 𝕂 = ℍ. The commutators in H_{ℍ}^{4n} are given by [(x, 0), (y, 0)] = (0, Pu ((x|y))) for some anisotropic hermitian form (·|·) on ℍ^{n}. We consider a pure quaternion h ∈ Pu (ℍ). As before, we have S ∩ hS ≠ {0}, for dimension reasons, and find u, v ∈ V such that vφ̄ = huφ̄. Computing [(v, 0), (u, 0)]^{φ} = (0, Pu ((vφ̄|uφ̄))) = (0, h(uφ̄|uφ̄)) ∈ (0, h)ℝ we find that the commutator of (H_{𝔽}^{2k})^{φ} ≅ H_{𝔽}^{2k} contains {0} × Pu (ℍ), and 𝔽 = ℍ.

Theorem 3.7. There is an embedding of H_{3}^{3} into H_{ℍ}^{4} .

Proof. We construct the symplectic map β: ℝ^{4} × ℝ^{4} → ℝ^{3} corresponding to H_{ℍ}^{4} , as follows: identify the ℝ[SO_{4}ℝ]-module ℝ^{4} ∧ ℝ^{4} with the Lie algebra 𝔬_{4} , and let β̄ be the projection onto some 3-dimensional ideal Z.

Now restrict the ℝ[SO_{4}ℝ]-module ℝ^{4} to an ℝ[SO_{3}ℝ]-module, and let S be the (unique) submodule of dimension 3. Then the submodule S ∧ S is not invariant under the action of the larger group SO_{4}ℝ. Therefore, it is not one of the ideals, and its image under β̄ is a nontrivial ℝ[SO_{3}ℝ]-submodule of Z. This means (S ∧ S)β̄ = Z, and we have found the subgroup GH (S, Z, β|_{S×S}) ≅ H_{3}^{3} inside GH (ℝ^{4}, Z, β) ≅ H_{ℍ}^{4} .

Theorem 3.8. There are embeddings of H_{ℂ}^{4} into H_{6}^{6} and of H_{ℍ}^{4} into H_{7}^{7} . However, there are no injective continuous homomorphisms from H_{ℝ}^{4} or H_{ℍ}^{4} into H_{6}^{6} , and no injective continuous homomorphisms from H_{ℝ}^{4} or H_{ℂ}^{4} into H_{7}^{7} .

Proof. For z ∈ {6, 7}, we model H_{z}^{z} by the group GH (ℝ^{z}, Z, β_{z}) obtained from a suitable projection β̄_{z}: 𝔬_{z} → Z, where 𝔬_{z} is considered as an ℝ[Δ]-module with Δ ∈ {SU_{3}ℂ, G_{2}}, according to the value of z.

Guided by 3.2, we search for a vector subspace S of ℝ^{z} such that Y := (S∧S)β̄_{z} has dimension 1, 2 or 3. The restriction γ: S×S → Y of β_{z} then gives a Heisenberg group GH (S, Y, γ). We will find S together with a group of automorphisms of GH (S, Y, γ) in such a way that we also see that GH (S, Y, γ) is almost homogeneous; together with the parameters, this determines GH (S, Y, γ) up to isomorphism.

Let S < ℝ^{z} be a vector subspace of dimension 4. Then, under our usual identification of ℝ^{z} ∧ ℝ^{z} with 𝔬_{z}, the subspace S ∧ S corresponds to the Lie algebra of the pointwise stabilizer of the orthogonal complement S^{⊥} of S in ℝ^{z}. This allows to determine 𝔨 := (S ∧ S) ∩ ker β̄_{z}, as follows.

For z = 6 we have that ker β̄_{6} is the Lie algebra of U_{3}ℂ, and 𝔨 either is the Lie algebra of U_{2}ℂ (if S is a complex subspace), or of U_{1}ℂ. In the first case, we obtain dim Y = 2, and the action of U_{2}ℂ shows GH (S, Y, γ) ≅ H_{ℂ}^{4} . The second case, where dim Y = 5, does not lead to an almost homogeneous subgroup.

For z = 7 we have to consider stabilizers in the group G_{2} . We regard ℝ^{7} as the space Pu (𝕆) of pure octonions in such a way that G_{2} acts as the group Aut(𝕆). If S^{⊥} generates 𝕆, the pointwise stabilizer of S^{⊥} in Aut(𝕆) is of course trivial. This means dim Y = 6, and we do not obtain an almost homogeneous group. The situation is different if S^{⊥} generates a proper subalgebra of 𝕆: this subalgebra is isomorphic to ℍ, and the pointwise stabilizer of S^{⊥} in Aut(𝕆) is a group isomorphic to U_{1}ℍ ≅ SU_{2}ℂ, while a subgroup Σ ≅ SO_{4}ℝ of Aut(𝕆) leaves S^{⊥} and S invariant, compare [3] 11.31. This yields dim Y = 3, and the action of Σ shows that GH (S, Y, γ) is almost homogeneous.

Theorem 3.9. There is an embedding of H_{ℝ}^{4} into H_{5}^{8} .

Proof. We consider H_{5}^{8} = GH (ℍ^{2}, Z, β) as in 2.3: the symplectic map β is obtained from the projection of the ℝ[U_{2}ℍ]-module ℍ^{2} ∧ ℍ^{2} onto a fivedimensional submodule Z.

We claim that S := ℍ × {0} generates a subgroup GH (S, Y, γ) ≅ H_{ℝ}^{4} in H_{5}^{8} where, of course, we have γ := β|_{S×S}. We know that S ∧ S is the Lie algebra of the pointwise stabilizer Γ of S^{⊥} = {0}×ℍ in SO_{8}ℝ. The intersection Δ := Γ ∩ U_{2}ℍ is a group isomorphic to U_{1}ℍ. As the central involution of U_{2}ℍ is not contained in Δ, we have that Δ acts effectively on Z, and the ℝ[Δ]- module Z splits as the sum of F := Fix_{Z}(Δ) and some irreducible module of dimension 4. The central involution of Δ acts trivially on Y because it induces −id on S. Thus Y is contained in F. In order to see that equality holds, we consider the projection π from the ℝ[U_{2}ℍ]-module 𝔬_{8}ℝ onto the complement C of the Lie algebra 𝔤 of 𝕊_{ℍ} · U_{2}ℍ. The centralizer of Δ in 𝕊_{ℍ} · U_{2}ℍ has dimension 6. Since Δ has 9-dimensional centralizer in SO_{8}ℝ, this implies that (Fix_{𝔬_{8}}(Δ)) π = Fix_{C}(Δ) has dimension 3. The kernel of the restriction π|_{S∧S} is the intersection of S ∧ S with 𝔤, and thus 3-dimensional. We obtain the equalities (S ∧ S)π = Fix_{C}(Δ) and (S ∧ S)β̄ = F.

I did not find any embeddings from H_{𝔽}^{4} into H_{5}^{8} , for 𝔽 ∈ {ℂ, ℍ}. According to 3.2, the only other almost homogeneous Heisenberg group admitting an embedding into H_{5}^{8} would be H_{ℝ}^{6} .

The main problem is that the ℝ[U_{2}ℍ]-module 𝔬_{8} has lots of submodules: it splits as the sum of the 10-dimensional Lie algebra of U_{2}ℍ, its 3-dimensional centralizer, and the 15-dimensional sum of (pairwise isomorphic, and thus not uniquely determined) 5-dimensional modules. The search for embeddings of H_{𝔽}^{4} (or for proofs of nonexistence of such embeddings) would have to consider subalgebras S ∧ S ≅ 𝔬_{4} corresponding to 4-dimensional subspaces S of ℝ^{8} . For such subalgebras, one needs to know the intersection with the kernel of the projection onto one of the 5-dimensional submodules.

For all (v, z) ≠ (8, 5), the exceptional almost homogeneous Heisenberg group H_{z}^{v} is described by a rather simple factorization of the module 𝔬_{v}. We use this in the sequel to obtain a fairly complete understanding of embeddings.

Theorem 3.10. There are no injective continuous homomorphisms from H_{ℝ}^{6} or H_{6}^{6} into H_{7}^{7} .

Proof. Let V = Pu (𝕆) = Z, and define β: V × V → Z by (v, w)^{β} := [v, w] := vw − wv. Then H := GH (V, Z, β) = C_{𝕆} ≅ H_{7}^{7} . Assume that G < H is a subgroup satisfying G ≅ H_{ℝ}^{6} or G ≅ H_{6}^{6} . From 3.2a we know G′ = H′ ∩ G. The commuting diagram

[figure: H/H′ × H/H′; β; H′; G/G′ × G/G′; β; G′]

shows that there is a hyperplane T of Pu (𝕆) ≅ H/H′ such that dim[T, T] = 1 or dim[T, T] = 6, respectively. Every hyperplane in Pu (𝕆) is the orthogonal space x^{⊥} of some nonzero element x ∈ Pu (𝕆). Using transitivity properties of Aut(𝕆), we may assume T = i^{⊥}. Now we compute 2i = [j, k] ∈ [i^{⊥}, i^{⊥}] and −2j = [jl, l] ∈ [i^{⊥}, i^{⊥}]. The orbit of j under the stabilizer of i in Aut(𝕆) is the unit sphere in i^{⊥}, see [3] 11.30. Therefore, we obtain Pu (𝕆) = [i^{⊥}, i^{⊥}], contradicting our assumptions. This shows that there is no subgroup isomorphic to H_{ℝ}^{6} or H_{6}^{6} in H_{7}^{7} .

Theorem 3.11. There is an embedding of H_{6}^{6} into H_{6}^{8} , but there is no continuous injective homomorphism from H_{ℝ}^{6} into H_{6}^{8} . Proof. Choose a vector subspace S of ℝ^{8} , and assume that dim S = 6. Then S ∧ S corresponds to the Lie algebra 𝔤 of a subgroup Γ ≅ SO_{6}ℝ in SO_{8}ℝ, and Γ∩U_{4}ℂ ≅ U_{3}ℂ (if ℂS = S) or Γ∩U_{4}ℂ ≅ U_{2}ℂ (if S is not a complex subspace). The ℝ[SU_{4}ℂ]-module 𝔬_{8} has two 6-dimensional simple submodules Z_{1} and Z_{2} whose sum Z_{1}+Z_{2} forms a complement to the Lie algebra 𝔲 of U_{4}ℂ. Restricting the projection π: 𝔬_{8} → Z_{1} +Z_{2} modulo 𝔲 to the 15-dimensional space S ∧S = 𝔤, we find dim(𝔤 ∩ 𝔲) ∈ {9, 4}. In the latter case, the projection of 𝔤 to Z_{l} has dimension at least 11 − 6 = 5. Surely, this does not yield an embedding of H_{ℝ}^{6} .

It remains to consider the case where S is a complex subspace. Then there is at least one simple ℝ[SU_{4}ℂ]-submodule Z of Z_{1} + Z_{2} such that the projection onto this submodule has nontrivial restriction β̄ to S ∧ S. As an ℝ[SU_{3}ℂ]-submodule, the module Z has to be simple (note that all the trivial ℝ[SU_{3}ℂ]-submodules of 𝔬_{8} are contained in 𝔲). We have thus found a subgroup GH (S, Z, β) of H_{6}^{8} such that GH (S, Z, β) ≅ H_{6}^{6} .

Theorem 3.12. There is an embedding of H_{7}^{7} into H_{7}^{8} . Moreover, up to automorphisms of H_{7}^{8} , any almost homogeneous Heisenberg group properly contained in H_{7}^{8} is contained in the image of this embedding. Consequently, there are no continuous injective homomorphisms from H_{ℝ}^{4} , H_{ℂ}^{4} , H_{ℝ}^{6} or H_{6}^{6} into H_{7}^{8} . Proof. The group Spin_{7} contains a subgroup isomorphic to G_{2} (this is just the stabilizer in the transitive action of Spin_{7} on 𝕊_{7} ). Restricting the simple ℝ[Spin_{7}]-module ℝ^{8} to an ℝ[G_{2}]-module, we obtain a (unique) submodule S of dimension 7. Under the identification of ℝ^{8} ∧ ℝ^{8} with 𝔬_{8} , the exterior product S ∧ S corresponds to the Lie algebra of the stabilizer of a vector. Thus it does not coincide with the Lie algebra of Spin_{7} . This means that the projection β̄ from the ℝ[Spin_{7}]-module 𝔬_{8} onto its 7-dimensional submodule Z (leading to the Heisenberg group GH (ℝ^{8}, Z, β̄) ≅ H_{7}^{8} ) restricts to a nontrivial map from S ∧ S to Z. But the group G_{2} acts irreducibly on Z, and we obtain that GH (S, Z, β|_{S×S}) is a subgroup of GH (ℝ^{8}, Z, β̄) with the property GH (S, Z, β|_{S×S}) ≅ H_{7}^{7} .

Now assume that H < H_{7}^{8} is an almost homogeneous Heisenberg group. Then H ≠ H_{7}^{8} implies H + Z ≠ H_{7}^{8} , see 3.2d. Thus (H + Z)/Z is a proper subspace of the vector space H_{7}^{8}/Z, and its orthogonal complement contains a nonzero vector v. As the orbit of v under Spin_{7} meets the orthogonal complement of (S + Z)/Z, we may assume (H + Z)/Z ≤ (S + Z)/Z, and H ≤ S + Z, as required. The rest follows from 3.8 and 3.10.

Open Problems. It remains to answer the following questions:

(a) Is there an embedding of H_{ℝ}^{4} into H_{6}^{8} ?

(b) Is there an embedding of H_{ℂ}^{4} into H_{5}^{8} ?

(c) Is there an embedding of H_{ℍ}^{4} into H_{5}^{8} , or into H_{6}^{8} ?

(d) Is there an embedding of H_{ℝ}^{6} into H_{5}^{8} ?

Note that in each of the open cases the larger group has a dimension which is considerably larger than that of the subgroup in question. Apart from these special problems, there are the following, more general ones:

(e) For which pairs (k, n) of positive integers is there an embedding of H_{ℂ}^{4k} into H_{ℍ}^{4n} ? (Partial answers have been given in 3.4, 3.5, and 3.6.)

(f) In those cases where embeddings exist: what can be said about uniqueness?

The following diagram attempts to visualize the embeddings of almost homogeneous Heisenberg groups, as far as we have found them. The diagram contains all almost homogeneous Heisenberg groups of dimension at most 18; in particular, all exceptional ones. Absence of (paths along) arrows indicates that no embedding exists. There remain the Open Problems, as stated above (and indicated by dotted arrows in the diagram). See, however, the results in 3.3, 3.8, 3.10, and 3.11.

Acknowledgements. The present investigation has benefitted from discussions with Sven Boekholt. The commutative diagrams have been produced with the help of Paul Taylor’s package (available by anonymous ftp from

ftp://ftp.dcs.qmw.ac.uk/pub/tex/contrib/pt/diagrams/ ).

References

[1] Bröcker, T., and T. tom Dieck, “Representations of compact Lie groups,” Graduate Texts in Math. 98, Springer, New York, etc., 1985.

[2] Mäurer, H., and M. Stroppel, Groups that are almost homogeneous, Geometriae Dedicata 68 (1997), 229–243.

[3] Salzmann, H., D. Betten, T. Grundhöfer, H. Hähl, R. Löwen, and M. Stroppel, “Compact projective planes,” Expositions in Mathematics 21, De Gruyter, Berlin etc., 1995.

[4] Stroppel, M., Homogeneous symplectic maps and almost homogeneous Heisenberg groups, Forum Math. 11 (1999), 659–672.

[5] Stroppel, M., Locally compact groups with many automorphisms, Manuscript, Stuttgart, 1999.

Markus Stroppel Mathematisches Institut B, Universität Stuttgart, D-70550 Stuttgart, Germany

Received September 17, 1999 and in final form December 1, 1999